Two-Dimensional Motion Projectile Motion Periodic Motion.

Two-Dimensional Motion

Projectile MotionProjectile Motion

Periodic MotionPeriodic Motion

Projectile MoionVx

Vx

Vx

Vy

Vy

Vy

Vx = constant

Vy = varying

Vx

Vx

Vx

Vy

Vy

Vy

Formulas:

Vx = constant therefore,

Vx = d/t

Vy = varying therefore, acceleration

vf = vi + atvf

2 = vi2 + 2ad

d = vi + 1/2at2

Projectile Motion

vi vy

vx

Vy = sin(vi)

Vx = cos(vi)

Vy controls how long it’s in the air and how high it goes

Vx controls how far it goes

Projectile Motion“Range formula”

vi R = vi2 sin2/g

yiyfRange formula works only when yi = yf

Remember!!!!! vi is the velocity at an angle and the sin2 is the sine of 2 x

Projectile Motion“Range formula”

vi

R = vi2 sin2/g

yiyf

If vi = 34 m/s and is 41o then,

R = 1160 m2/s2 (0.99)/9.8 m/s2

R = 120 m

R = (34 m/s)2 sin82o/9.8 m/s2

Projectile Motion“Range formula”

vi

Note that if becomes the complementof 41o, that is, is now 49o, then,

vi = 34 m/s and is 49o then,

R = 1160 m2/s2 (0.99)/9.8 m/s2

R = 120 m

R = (34 m/s)2 sin98o/9.8 m/s2

So, both 41o and 49o yield “R”

Projectile Motion“Range formula”

vi

yiyf

OR,

If vi = 34 m/s and is 41o then,vyvy = sin41o(34m/s) = 22m/s, and

t = vfy - viy/g = -22m/s - (22m/s)/-9.8m/s2 = 4.5 s

vx

dx = vx(t) = 26m/s (4.5 s) = 120 m

vx = cos4 m/s) = 26 m/s, and

Circular Motion

When an object travels about a When an object travels about a given point at a set distance it is said given point at a set distance it is said to be in to be in circular motioncircular motion

Cause of Circular Motion

11stst Law…an object in motion stays in motion, in Law…an object in motion stays in motion, in

a straight line, at a constant speed unless acted a straight line, at a constant speed unless acted on by an outside force.on by an outside force.

22ndnd Law…an outside force causes an object to Law…an outside force causes an object to accelerate…a= F/maccelerate…a= F/m

THEREFORETHEREFORE, circular motion is caused by a , circular motion is caused by a force that causes an object to travel contrary to force that causes an object to travel contrary to its inertial pathits inertial path

Circular Motion Analysisv1

v2

rr

Circular Motion Analysis

v1

v2

rr

v1

v2

v = v2 - v1

or v = v2 + (-v1)

(-v1) = the opposite of v1

v1

(-v1)

v1

v2

rr0 v = v2 - v1

or v = v2 + (-v1)

(-v1) = the opposite of v1

v1

(-v1)

v1

v2

v2

(-v1)

vNote how v is directed toward the center of thecircle

v1

v2

rr

v1

v2

v2

(-v1)

v

Because the two triangles aresimilar, the angles are equal andthe ratio of the sides areproportional

l

v1

v2

rr

v1

v2

v2

(-v1)

v

l

Therefore,

v/v ~ l/r and v = vl/r

now, if a = v/t, and v = vl/r

then, a = vl/rt, since v = l/t

THEN, a = v2/r

Centripetal Acceleration

ac = v2/r

now, v = d/t and, d = c = 2r

then, v = 2r/t and, ac = (2r/t)2/r

or, ac = 42 r2/t2/r or, ac = 42r/T2

The 2nd Law and Centripetal Acceleration

Fc

ac

vt

F = ma

ac = v2/r = 42r/T2

therefore,

Fc = mv2/r or,

Fc = m42r/T2

Simple Harmonic Motionor S.H.M.

Simple Harmonic motion is motion that has force and acceleration always directed toward the equilibrium positionand has its maximum values when displacement is maximum.Velocity is maximum at the equilibrium position and zero atmaximum displacement

Pendulum motion, oscillating springs (objects), and elastic objects are examples

F = maxa = maxv = 0

F = maxa = maxv = 0

F = greatera = greaterv = lessF = 0

a = 0v = max

F = lessa = lessv = greater

Simple Harmonic Motion

Force

acceleration

Pendulum Motion

Fw

FT

Note that FT (the accelerating forceis a component of the weight of thebob that is parallel to motion (tangentto the path at that point).

Pendulum Motion

Fw

FT

Note that as the arc becomes lessso does the FT, therefore the force and resulting acceleration also becomes less as the “bob” approaches the equilibrium position.

Pendulum Motion

Fw

FT

ac = r/T2

ac = g and r = l

g = l/T2

T2 = l/g

T = 2 l/g

Oscillating Elastic ObjectsFe = max Fe = maxFe = less Fe = less

a = max a = maxa = lessa = less

a and F = 0

Fw

FT

Note that no part of Fw is in thedirection on Motion, or FT

There, F and a is zero!!!